D is the mid-point of side BC of Δ ABC and E is the mid-point of BD.

1.	D is the mid-point of side BC of Δ ABC and E is the mid-point of BD. If O is the mid-point of AE, Prove that, ar(ΔBOE) = \(\frac{1}{8}\)ar(ΔABC)
Answer» Join A and D to get AD median (Median divides the triangle into two triangles of equal area) Therefore, Area(ΔABD) = \(\frac{1}{2}\)Area (ΔABC) Now, Join A and E to get AE median Similarly, We can prove that, Area(ΔABE) = \(\frac{1}{2}\)Area (ΔABD) Area(ΔABE) = \(\frac{1}{4}\)ABC (Area (ΔABD) = \(\frac{1}{2}\)Area (ΔABC) (i) Join B and O and we get BO median Now, Area(ΔBOE) = \(\frac{1}{2}\)Area (ΔABE) Area(ΔBOE) = \(\frac{1}{2}\) x \(\frac{1}{4}\)Area (ΔABC) Area(ΔBOE) = \(\frac{1}{8}\)Area (ΔABC)

D is the mid-point of side BC of Δ ABC and E is the mid-point of BD. If O is the mid-point of AE, Prove that, ar(ΔBOE) = \(\frac{1}{8}\)ar(ΔABC)

Answer»

Join A and D to get AD median

(Median divides the triangle into two triangles of equal area)

Therefore,

Area(ΔABD) = \(\frac{1}{2}\)Area (ΔABC)

Now,

Join A and E to get AE median

Similarly,

We can prove that,

Area(ΔABE) = \(\frac{1}{2}\)Area (ΔABD)

Area(ΔABE) = \(\frac{1}{4}\)ABC (Area (ΔABD) = \(\frac{1}{2}\)Area (ΔABC) (i)

Join B and O and we get BO median

Now,

Area(ΔBOE) = \(\frac{1}{2}\)Area (ΔABE)

Area(ΔBOE) = \(\frac{1}{2}\) x \(\frac{1}{4}\)Area (ΔABC)

Area(ΔBOE) = \(\frac{1}{8}\)Area (ΔABC)